where \(N\)  is the dimension of the momentum space of the PTKR model, and \(\theta _c=\pi /2\) .
The above prediction is verified by numerical results in Fig. REF . As an illustration, we consider \(m=1\) , 2, and 3 in numerical simulations. Figure REF (a) shows that for a specific \(m\) , the \(C(t)\)  saturates rapidly as time evolves, which is in perfect agreement with our theoretical prediction in Eq. (REF ). In order to further confirm the scaling law of \(C(t)\) , we numerically investigate the \(C\)  at a specific time for different \(N\) . Figure REF (b) shows that for \(t=t_{10}\) , the value of \(C\)  increases in the power-law of \(N\) , which coincides with the theoretical prediction in Eq. (REF ). The scaling of \(C(t_{})\)  with dimension of the system demonstrates that it diverges as \(N\rightarrow \infty \) , which is of interest in the study of fast scrambling [1]}. We would like to mention that we previously found the scaling law for the OTOCs constructed by \(A=\theta \)  and \(B=p\) , in a Gross-Pitaevski map system [2]}. Our present work explores the scaling law for \(B=p^m\)  with arbitrary integer \(m\) , moreover extends the investigation to non-Hermitian systems, which is obviously an significant advance in the fields of operator growth in chaotic systems.
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